On the compactness of a class of Riemannian manifolds
A compactness property is proved for the class \({\mathcal L} = {\mathcal L} (H,K, V, n, i_0)\) of \(n\)-dimensional Riemannian manifolds \((M,g)\) satisfying the following conditions: a) \(M\) is diffeomorphic to the standard Euclidean ball \(B_2 (0)\) of radius 2; b) \(|(\text{Ric }g) (x)|\leq Hr^{-2}\), where \(r = \text{dist} (x,0)\); c) the injectivity of \((M,g)\) is bounded below by \(i_0 > 0\); d) \(\int_M |\text{Rm} (g) |^{n/2} dg \leq K\), where \(|\text{Rm}(g)|\) denotes the norm of the Riemann curvature tensor of the metric \(g\); e) \(\text{volume}(M,g) \leq V\). The main result asserts that for any sequence of \(\mathcal L\) there exists a subsequence \(\{(M_k, g_k)\}\), a \(C^\infty\)-manifold \(M'\) diffeomorphic to \(B_2 (0)\) and a \(C^0\)-metric \(g'\) on \(M'\) such that \(g_k \to g'\) in the \(C^0\)-norm on \(M'\), and the convergence is in the \(C^{1,\alpha}\)-norm away from 0. As an application, the compactness of orbifolds with a finite number of singularities is discussed.
scientific article; zbMATH DE number 759232